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Proposition 21.12.3855BFC442EA72

Proposition 21.12.38.55BFC442EA72 Let $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$. Then the following are equivalent:

  1. 323127E50410

    $\varphi \in \mathcal{E}(X,\omega )$;

  2. BCA447CA7659

    $F_{\varphi }\in \mathcal{E}(N_{\mathbb {R}},P)$;

  3. E488FC604B8A

    $\overline{\operatorname{Dom}G_{\varphi }}=P$.

Proof. 1 $\iff $ 3. By proposition 21.12.24,

\[ \int _ X \omega _{\varphi }^ n=\int _{T(\mathbb {C})} \left(\omega |_{T(\mathbb {C})}+\mathrm{dd}^{\mathrm{c}}\varphi |_{T(\mathbb {C})} \right)^ n= n! \operatorname{vol}\overline{\operatorname{Dom}G_{\varphi }},\quad \int _ X \omega ^ n=n!\operatorname{vol}P, \]

where in the first equality, we use the fact that $\omega _{\varphi }^ n$ does not charge pluripolar sets proposition 13.11.4 4.

Therefore, 1 and 3 are equivalent.

2 $\iff $ 3. This follows from proposition 21.11.34. $\square$

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